Consider the linear dynamical system

that is

We look for solutions on the form

If we instert these to the system we get

that is

This equation system has nontrivial solutions if and only if

that is, if and only if

This is the *characteristic equation* of the system and the solutions its *eigenvalues*. If we now for each eigenvalues solve the equation system

we obtain the eigenvectors

Then we have the general solution of our original dynamical system as

that is

where *C _{1}* and

Example 8:

This matrix has characteristic equation

with the eigenvalues

and corresponding eigenvectors

This means that the general solution of the corresponding dynamical system is

that is

**Solution:** We consider the matrix

and its characteristic equation

that is

The eigenvalues thus are

with corresponding eigenvectors

This means that the dynamical system has the general solution

that is

These are all *complex *solutions. We are actually only interested in the *real*solutions. With help of Euler’s formula we get

If we now pick arbitrary real constants *D _{1}* and

we get the general real solution